GCSE Maths Congruence & transforms revision

Two shapes are congruent when one fits exactly on the other — equal sides and equal angles throughout. Triangles can be proved congruent from just three carefully chosen facts: SSS (all three sides), SAS (two sides with the angle *between* them), ASA (two angles with the *corresponding* side), and RHS (right angle, hypotenuse and one further side). The angle in SAS must sit between the two sides, and the side in ASA must correspond to the pair of angles — otherwise the facts pin down no unique triangle. “SSA” (angle not included) is the famous fake criterion: two different triangles can share it.

Congruence statements respect letter order: writing △ABC ≅ △PQR claims A ↔ P, B ↔ Q, C ↔ R, so side AB must equal PQ. Exam questions often bury the third fact (a shared side, a pair of vertically opposite angles, or parallel lines giving equal alternate angles) — hunt for it before naming the criterion.

Similar shapes keep their angles and stretch all lengths by one length scale factor k. Find k from any known pair (image ÷ original, keeping corresponding sides), then multiply or divide the missing lengths. Triangles 6–4–? enlarged so the side matching 6 becomes 9 have k = 9 ÷ 6 = 1.5, so the side matching 4 becomes 6. Perimeters scale by k too — a perimeter is a length.

Areas and volumes obey their own powers of k (Higher): doubling every length quadruples the area (k² = 4) and multiplies the volume by 8 (k³). Going backwards — original from image — divides: image area 45 with k = 5 came from 45 ÷ 25 = 1.8. The commonest exam trap is using k instead of k² for area; test yourself by asking “do I count this shape in one-dimensional strips, in squares, or in cubes?”

A full description leaves nothing to guess. Translation: give the column vector — positive x right, positive y up. Reflection: name the mirror line exactly (“the y-axis”, “y = x”), never “the diagonal”. Rotation: give centre, angle *and* direction (clockwise or anticlockwise). Enlargement: give the centre and scale factor. Tracing paper is allowed for rotations and reflections in the exam — use it, then state the description in words.

Enlargement with k > 1 grows and 0 < k < 1 shrinks; measure every image distance from the *centre*, along the ray through each vertex. A negative scale factor (Higher) places the image on the opposite side of the centre: with centre O and k = −2, the point (2, 0) maps to (−4, 0). Invariant points survive the move untouched: every point on a reflection line, and the centre of a rotation or enlargement. Under a rotation about O, only O itself stays fixed.

A column vector ( x over y ) means “x right, y up”. Arithmetic is component-wise: (4, −2) + (−1, 5) = (3, 3), and multiplying by a scalar hits both components: 2 × (−3, 4) = (−6, 8). Chained journeys join head to tail: going A to B then B to C equals going A to C directly — AB + BC = AC. Reversing a journey flips the sign: BA = −AB.

Proofs (Higher) lean on two facts: parallel vectors are scalar multiples of each other, and a shared magnitude plus direction proves equality. In parallelogram OACB with OA = a and OB = b, the side AB is b − a. The midpoint M of AB sits at a + ½(b − a) = ½(a + b) — exactly half of OC = a + b, so M is also the midpoint of the diagonal OC: the diagonals bisect each other. That chain — express, simplify, interpret — is the whole of vector proof.